Abstract
This book is devoted to the study of Euclidean Geometry , so named after the famous book Elements , of Euclid of Alexandria .
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Notes
- 1.
Euclid of Alexandria , Greek mathematician of the fourth and third centuries bc, and one of the most important mathematicians of classical antiquity. The greatest contribution of Euclid to Mathematics, and to science in general, was the treatise Elements, in which he systematically exposed all knowledge of his time in Geometry and Arithmetic. The importance of the Elements lies in the fact that it was the first book ever written in which a body of mathematical knowledge was presented in an axiomatic way, with all arguments relying solely on Logic.
- 2.
An axiom or postulate is a property imposed as true, without the need of a proof. The use of the axiomatic method is one of the most fundamental characteristics of Mathematics as a whole.
- 3.
For the time being, we also implicitly assume that all points under consideration are contained in a single plane, and that there exists at least three points not situated in the same line.
- 4.
This is the first one of a series of examples whose purpose is to develop, in the reader, a relative ability on the use of straightedge and compass. In all of them, we list a sequence of steps which, once followed, execute a specific geometric construction. After attentively reading each of these examples, we strongly urge the reader to reproduce the listed steps by him/herself to actually execute the geometric constructions under consideration. Finally, and for the sake of rigor, we observe that a geometric construction does not constitute a proof of a geometric property, for it necessarily involves precision errors and particular choices of positions. Its main purpose is to help developing geometric intuition.
- 5.
Of course, at this point we rely on the reader’s intuition, or previous knowledge, for the meaning of equal.
- 6.
Strictly speaking, the following argument is fallacious, for, among other things, it invokes the notion of length of an arc of a circle, something which has not yet been defined. Nevertheless, it develops quite a useful intuition for the measurement of angles that is enough for our purposes along these notes. For a thorough discussion of the measurement of angles, we refer the reader to [19].
- 7.
The exception will be the lower case greek letter π (one reads pi). As we shall see in Chap. 5, we will reserve π to denote the area of a disk or radius 1.
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Caminha Muniz Neto, A. (2018). Basic Geometric Concepts. In: An Excursion through Elementary Mathematics, Volume II. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-77974-4_1
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